Succinct representations of planar maps

Abstract

International audienceThis paper addresses the problem of representing the connectivity information of geometric objects, using as little memory as possible. As opposed to raw compression issues, the focus here is on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose, in particular, the first optimal representations for 3-connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3-connected planar graphs, and 1.62 bits per triangle, or equivalently 3.24 bits per vertex for triangulations. These representations support adjacency queries between vertices and faces in constant time

Catalog-Based Representation of 2d Triangulations

“…In this paper, we have proposed a new representation for triangulations inspired from previous theoretical work [6]. We describe several catalogs of tiny patches that are stable under relevant update operations which can be used as patterns to recognize in the triangulation.…”

“…This allows us to avoid representing the details of the connectivity inside a patch (since they are common to all patches of the same kind, and thus represented only once). The previous work [6] was theoretically optimal but quite far of a practical application since it was very complicated to implement and since the asymptotic optimal behavior is reached for highly unrealistic size of triangulation. This work gives practical analysis and experimental evidence that a simplified version of this work allows one to reduce significantly the storage needed allowing a trade-off between time and space needed by a triangulation.…”

“…In previous papers, we have proposed an optimal way of representing a triangulation of n points using 3.24 n bits [5,6], with an additional storage which is asymptotically negligible (in the case of a triangulation of a topological sphere; for the triangulation bounded by a polygon of arbitrary size the cost is 2.17 bits per triangle). The idea is to gather triangles in tiny patches of size between log n 12 and log n 4 , and to introduce a graph of patches to describe adjacency relations between them.…”

“…. , t p } of planar triangulations with a simple boundary of arbitrary size, called patches (the t i are called tiny triangulations in previous papers [5,6]). A stable catalog for a given operation is a catalog, where the result of each update operation applied to a triangulation formed by one or several patches of the catalog is either included in the catalog, or decomposable (within a restricted neighborhood) to some elements of the catalog.…”

Vertices of Degree k in Random Maps

“…First, Fusy, Poulalhon, and Schaeffer [10] discovered a beautiful bijection between the class of cnets and a class of plane trees, which not only provides a combinatorial interpretation of the formula enumerating c-nets with a given number of vertices and faces, but it also solves the problem of compressing efficiently the connectivity information encoded in such a map. Moreover, Castelli, Aleardi, Devillers, and Schaeffer [4] gave, among other results, optimal representations for some classes of maps and triangulations. These are of particular significance, since triangulations are the most standard classes of graphs underlying meshes with spherical topology.…”